Referring to FIG. 1A, a circuit, known as the Chua circuit 1, oscillates chaotically. The term “chaos” applies to dynamic systems that follow simple dynamical rules, but whose state function trajectory is so sensitive to the system's initial conditions that its state after an arbitrary time-period cannot, in practical terms, be predicted. That is, its state could be predicted if it were possible to model the system with an arbitrary degree of precision. Chaotic systems evolve deterministically, and, their chaotic state paths are cyclic, but very complex and with extremely long cycle-lengths. In real systems, however, with extremely long cycle periods, it may be of little practical significance that their behavior is cyclical because the physical systems that generate the behavior may not be sufficiently stable for the system to ever return to the same dynamical system in its same initial state. For example, the component values of an electrical circuit may not remain precisely constant for 600 years.
The Chua circuit, a simple electrical circuit that exhibits chaotic behavior. It has been studied extensively and used to demonstrate many of the chaotic patterns observed in many physical systems. Referring now also to FIG. 1C, the basic Chua circuit includes a non-linear resistance element 10, characterized by a non-linear voltage-current characteristic curve. In a typical configuration, the curve is piece-wise linear with symmetrical slope discontinuities around the zero-axis. That is: IR=GaVR+(½)(Ga−Gb){|vR+Bp|−|vR−Bp|} where Ga and Gb are the slopes of respective linear portions of the piecewise-linear current/voltage curve characterizing the non-linear resistor and Bp is the absolute value of the two voltage points at which the discontinuities in the current/voltage curve lie as shown in FIG. 1C. The circuit has a circuit-driving subsystem 2, a L-C tank circuit, and a response subsystem 3, with a capacitance/non-linear resistor, interconnected through a resistor 25.
Referring to FIG. 1D, a given choice of values of the physical characteristics of the components of the Chua circuit each correspond to a unique operating regime, some values of which may coincide with a chaotic behavior of the Chua circuit. The operating regime may be mapped onto a coordinate system whose axes are the lump parameters, α=C2/C1 and β=R2C2/L. By choosing values of R, L, C1 and C2 so that α and β lie in, for example, a double scroll region 60, a Chua circuit can be made that will oscillate chaotically or quasi-periodically. The points on the plot correspond to different operating behaviors and a given point does not exhaustively define a particular path of state trajectories. Some points may correspond to radically different behaviors depending on the initial conditions. Given a specified physical configuration and a specified initial state specified by V1, V2, and IL, the voltages across C1 and C2 and the current through L, the evolution of the Chua circuit's state is deterministic, but chaotic. That is, any Chua circuit with the same physical parameters and initial conditions will follow the same course of states over time and this course will repeat itself over a very long interval (perhaps many years). However, to an observer, the value of, say, V1 over a period of time shorter than this long interval, looks like noise. Also, initial states that differ only slightly can follow widely different state paths. In addition, its power spectral density function is spread over a wide range of frequencies, with a peak at the frequency of the fundamental of the L-C tank circuit formed by L and C2. However, compared to oscillators, such as used to generate carriers for radio transmission, the peak is not pronounced, that is, it is very short and wide.
The Chua circuit, aside from being a classic device for demonstrating, studying, and modeling chaotic real-world systems, has also been proposed as a basis for chaotic signal transmission. Generally a transmitting nonlinear dynamic circuit produces a chaotic signal that may be used to induce a receiving chaotic system to synchronize with it. The parameter of the transmitting chaotic circuit may be modulated or perturbed responsively to an information signal. The parameter may be a scalar, such as a voltage, tapped from the transmitting circuit and used as a signal. The signal is applied to the receiving system, causing the receiving system to synchronize with the transmitted signal. The chaotic signal from the synchronized receiving circuit may be used with the modulated transmitted signal to recover the information signal according to various prior art schemes. The chaotic signals derivable from an oscillating Chua circuit are similar to spread-spectrum signals including a range of frequencies. Chua circuits have been made to generate communications signals in frequency bands ranging from audio to radio frequency and in various media.
Various modulation schemes have been proposed. For example, a simple signal summing system adds the information signal to the chaotic scalar. A more complex correlation system uses a signal divider and multiplier at the transmitter and receiver, respectively. In FIG. 1B, a proposed transmitter and receiver design uses a Chua circuit to transmit signals and receive signals. The system has a transmitting Chua circuit 100 and an identical (in terms of its chaotic oscillating properties) receiving Chua circuit 101. The transmitting Chua circuit 100 oscillates in a chaotic or semiperiodic regime. Generally, the two chaotic circuits 100 and 101 may be synchronized by driving a portion of the receiving chaotic oscillator 101 with a driving function tapped from the transmitting chaotic oscillator 100. In the Chua circuit, a L-C tank circuit 105 of the transmitting Chua circuit 100 is linked through a resistor 81 to the capacitor/non-linear resistor portion 106. The latter portion causes the oscillations of the L-C tank circuit to become chaotic for certain values of the inductor 74, capacitors 71 and 73, and resistor 81 as discussed above with reference to FIG. 1D. The chaotic portion 108 of the identical receiving circuit 101, also a capacitor/non-linear resistor circuit, reproduces the driving signal. That is, the transmitting 100 and receiving 101 circuits follow precisely the same chaotic course of states (assuming no modulation is taking place in the transmitting circuit 100).
It is known that the transmitting 100 and receiving 101 circuits will remain synchronized even when a substantial amount of noise and/or information is injected into the driving signal. Thus, in the prior art embodiment of FIG. 1B, a signal current Ii(t) is injected by a driver 76 that converts a signal voltage through an invertable coding function c(vs(t)). The decoded signal at the receiver is then obtained from the received current signal Id(t) by applying the inverse coding operation to the received current signal Id(t) to obtain a voltage signal containing the information signal.
Note that the term, “synchronous,” in this context, characterizes the convergence of two state variables toward identical or linearly related, but continuously changing, sets of values. That is, a change in one variable corresponds to a change in a synchronized variable that is linearly related to the change in the one variable. Thus, plotting one variable against the synchronized variable over time, the result, theoretically, is a straight line. Synchronization of non-linear systems, and the mathematical modeling of such systems, is described in some detail in U.S. Pat. Nos. 5,245,660, 5,473,694, 5,402,334, 5,379,346 5,655,022, 5,432,697, and 5,291,555, the entirety of each of which is incorporated by reference herein.
Prior art systems have been discussed widely, but few practical working designs are known. The problems with practical synchronization systems are summarized in the introduction of U.S. Pat. No. 5,680,462. Synchronization systems are inherently noisy and error prone due, at least in part, to the time it takes for synchronization to occur in a noisy channel. For example, when a transmitting circuit is perturbed to encode a piece of information (a bit), it takes a finite amount of time for the receiving circuit to begin to follow the trajectory of the transmitted signal. Also, according to the prior art, modulation also cannot span too great a range. Otherwise, a tightly locked synchronization, which is, according to the prior art, essential, cannot be maintained. In addition, the practical problems attending achievement of high data throughput, the providing of reliable locking performance, and various purely practical design considerations have not received a great deal of attention. These prior art problems are addressed by the present invention.